Kepler's 3rd Law and the Doppler Effect

dekoi

Question:

Imagine a space probe has been placed in a circular orbit about a distant planet. The probe emits a continuous radio signal with a wavelength of 8 m. You measure the signal from earth, and find it to have a wavelength that varies regularly between 7.99943 m and 8.00057 m, with a period of 4.5 hours. Assuming that you are in the plane of the probe's orbit, and that you are not moving, calculate the mass of the planet.



This is what I have done...

By substituting into different equations, I end up with the equation:
a = λ / 2pi

Using Kepler's 3rd Law:

P^2 = a^3 / mt

I end up with :

mt= (λ / 2pi)^3(1 / P)^2

However, I do not know why I am given 3 different values for wavelength, should this be applied in to the answer or not?

SpaceTiger

You got that the semimajor axis of the orbit was comparable to the wavelength of the radiation? I'd like to see which equations you used to get that result. The only radiation for which that would be true is the gravitational variety.

Try instead using:

[tex]\frac{\Delta \lambda}{\lambda}=\frac{v}{c}[/tex]

and

[tex]v_c=\sqrt{\frac{GM}{a}}[/tex]

dekoi

I just did what my teaching assistant told me to do. I don't think it's correct either.

I'm assuming that "c" is the speed of light.

What is the M? Mass of the planet?

dekoi

And what can I do with that equation for v?

SpaceTiger

Sorry, I forgot to define my variables: M is the planet's mass, v is the velocity relative to your line of sight, v_c is the circular velocity of the orbit, [itex]\Delta \lambda[/itex] is the shift in wavelength relative to its rest frame value, [itex]\lambda[/itex] is the rest frame value of the wavelength, c is the speed of light, a is the semimajor axis, and G is the gravitational constant.

dekoi

We haven't learned that much information in order to use those equations. The most we have learned is Kepler's 3rd Law and the Gravitational force equation.

F=GMm/ R^2

SpaceTiger

The second equation follows simply from Kepler's Third Law, but you do have to know the first one in order to do the problem. I suggest a google (or PF) search on the doppler effect.

dekoi

Okay, thank you anyway.